Chen et al.'s Ore-degree conjecture for the strong chromatic index

Let $G$ be a finite simple connected graph. The Ore-degree of $G$, denoted by $\theta(G)$, is

$$\theta(G)=\max\{d(u)+d(v):uv\in E(G)\}.$$

Its strong chromatic index $\chi_s'(G)$ is the minimum number of colors in a strong edge-coloring, meaning that edges at distance at most $2$ in the line graph receive distinct colors.

Chen et al.'s conjecture. If $\theta(G)\ge 5$, then

$$\chi_s'(G)\le\begin{cases} 5\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil^2-8\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil+3 &\text{if } \theta(G)\equiv 1\mod 4,\\ 5\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil^2-6\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil+2 &\text{if } \theta(G)\equiv 2\mod 4,\\ 5\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil^2-4\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil+1 &\text{if } \theta(G)\equiv 3\mod 4,\\ 5\bigl\lceil\frac{\theta(G)}{4}\bigr\rceil^2 &\text{if } \theta(G)\equiv 0\mod 4.\end{cases}$$

This conjecture adapts the Erdős–Nešetřil bound by replacing maximum degree with Ore-degree. The case $\theta(G)=5$ was verified by Wu and Lin, but the parser marks the conjecture as resolved without supplying evidence that the full statement has been proved; its database status is therefore recorded as open pending verification.